Topics
6. Functions & Relations
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United States of America
Grade 8

6.05 Model linear relationships

Lesson
Practice

Introduction

Now that we know how:

  • to graph linear relationships
  • to find the equations of linear functions
  • to use algebra and graphs to extract information
  • to find intercepts and constant values, and
  • that the slope of a linear function represents constant change.

We can put this to use to solve a range of real life applications.

It's all the same mathematics, but this time you will have a context to apply it to.

Connections between representations

When given an equation in slope-intercept form, we can interpret the values of the slope and the y-intercept of a line.

When given a scenario or a table of values, we often will try to write a linear equation in slope-intercept form y=mx+b.

We may also use point-slope form to get the equation as well, y-y_1=m(x-x_1).

We can represent a linear relationship in many different ways including tables of values, graphs, or equations. The slope and y-intercept can be found from any representation.

Examples

Example 1

Consider the points in the table:

\text{Time in minutes } (x)12345
\text{Temperature in } \degree \text{C } (y)69121518
a

By how much is the temperature increasing each minute?

Worked Solution
Create a strategy

Refer to the table and find how much y increase for each 1 unit increase in x.

Apply the idea
\displaystyle \text{Temperature increase}\displaystyle =\displaystyle 9-6Find the difference between two temperatures
\displaystyle =\displaystyle 3\degree \text{C}Evaluate
b

What would the temperature have been at time 0?

Worked Solution
Create a strategy

Subtract the answer in part (a) from temperature after 1 minute.

Apply the idea
\displaystyle \text{Temperature}\displaystyle =\displaystyle 6-3Subtract 3 from the temperature at x=1
\displaystyle =\displaystyle 3\degree \text{C}Evaluate
c

Find the algebraic relationship between x and y.

Worked Solution
Create a strategy

Use the slope-intercept form with temperature increase in each minute as the slope and temperature at time 0 as the y-intercept to create an equation.

Apply the idea
\displaystyle y\displaystyle =\displaystyle mx+bWrite the slope-intercept form
\displaystyle y\displaystyle =\displaystyle 3x + 3Substitue m=3 and b=3
Idea summary

The slope and y-intercept can be found from all representations:

Table of values:

  • The slope is the constant rate of change. If the x-value is increasing by 1, then it will be the increase in the y-values.
  • The y-intercept is the value of y when x=0. This may not actually be given in the table, so you may have to count backwards using the rate of change.

Graph:

  • The slope can be read using the rise and run between two points.
  • The y-intercept is the value where the linear graph crosses the y-axis.

Equation:

  • In slope-intercept form, the slope will be the coefficient of the x-value. m in y=mx + b.
  • In slope-intercept form, the y-intercept will be the constant. b in y=mx + b.

When given a scenario or a table of values, we often will try to write a linear equation in slope-intercept form y=mx+b.

We may also use point-slope form to get the equation as well, y-y_1=m(x-x_1).

Interpret linear models

As mentioned above, we often use slope-intercept form to represent applications of linear relationships.

A common scenario is a cost function where there is both a fixed and variable component. For example, a plumber that charges a flat-fee of \$100 to show up plus an hourly rate of \$60. The initial or flat-fee would be the y-intercept, while the rate would be the slope.

We would get y=60x+100 as our equation.

Examples

Example 2

A carpenter charges a callout fee of \$150 plus \$45 per hour.

a

Write an equation to represent the total amount charged, y, by the carpenter as a function of the number of hours worked, x.

Worked Solution
Create a strategy

Use the slope-intercept form with the rate per hour as the slope and the call out fee as the y-intercept.

Apply the idea
\displaystyle y\displaystyle =\displaystyle mx+bWrite the slope-intercept form
\displaystyle y\displaystyle =\displaystyle 45x + 150Substitue m=45 and b=150
b

What is the slope of the function?

Worked Solution
Create a strategy

Refer to the value of m in the equation in part (a).

Apply the idea

m=45

c

What does this slope represent?

A
The total amount charged increases by \$45 for each additional hour of work.
B
The minimum amount charged by the carpenter.
C
The total amount charged increases by \$1 for each additional 45 hours of work.
D
The total amount charged for 0 hours of work.
Worked Solution
Create a strategy

The slope represents the rate of change of the total amount charged by the carpenter.

Apply the idea

From the equation created in part (a), we can say that the total amount charged increases by \$45 for each additional hour of work. Option A.

d

What is the value of the y-intercept?

Worked Solution
Create a strategy

The y-intercept is the value of y, when x=0.

Apply the idea
\displaystyle y\displaystyle =\displaystyle 45x+150Write the equation in part (a)
\displaystyle =\displaystyle 45\times0 + 150Substitute x=0
\displaystyle y\displaystyle =\displaystyle 150Evaluate
e

What does this y-intercept represent?

A
The total amount charged increases by \$150 for each additional hour of work.
B
The maximum amount charged by the carpenter.
C
The callout fee.
D
The minimum amount charged by the carpenter.
Worked Solution
Create a strategy

The y-intercept represents the total amount charged for 0 hours of work.

Apply the idea

We see that the value of y-intercept is 150 which is equal to the callout fee. Since it is the amount charged for 0 hours of work, we can say that it is the minimum amount charged by the carpenter. So options C and D are correct.

f

Find the total amount charged by the carpenter for 6 hours of work.

Worked Solution
Create a strategy

Substitute x=6 in the equation in part (a).

Apply the idea
\displaystyle y\displaystyle =\displaystyle 45\times6 + 150Substitute x=6
\displaystyle =\displaystyle 270+150Evaluate the multiplication
\displaystyle =\displaystyle \$420Evaluate
Idea summary

We often use slope-intercept form to represent applications of linear relationships. This means we use y=mx+b where m is the slope or rate of change and b is the y-intercept or initial value.

Use linear models

Once we have a linear model, we can use it to make predictions. If we substitute in for one variable we can use the linear equation to solve or evaluate for the other.

When you are substituting into a linear equation, be sure that you are substituting for the correct variable.

Examples

Example 3

The points show the relationship between water temperatures and surface air temperatures.

-4
-3
-2
-1
1
2
3
4
x
-10
-5
y
a

Complete the table of values:

\text{Water temperature } (\degree \text{C}), \, x-3-2-10123
\text{Surface Air Temperature }(\degree \text{C}), \, y
Worked Solution
Create a strategy

Identify the coordinates of the the points in the graph and fill the table with the y-values which are the Surface Air Temperatures corresponding to the x-values which are the Water Temperature.

Apply the idea
-4
-3
-2
-1
1
2
3
4
x
-10
-5
y
\text{Water temperature } (\degree \text{C}), \, x-3-2-10123
\text{Surface Air Temperature }(\degree \text{C}), \, y-14-11-8-5-214
b

Write an algebraic equation representing the relationship between the water temperature (x) and the surface air temperature (y).

Worked Solution
Create a strategy

Use the slope-intercept form to write an equation: y=mx+b.

Apply the idea

Find the slope (m) using the points (-3,-14) and (-2,-11):

\displaystyle m\displaystyle =\displaystyle \dfrac{y_2-y_1}{x_2-x_1}Write the slope formula
\displaystyle =\displaystyle \dfrac{-11-(-14)}{-2-(-3)}Substitute the values
\displaystyle =\displaystyle 3Evaluate

Looking at the table from part (a), the value of surface air temperature when water temperature is 0 is -5. So b=-5

\displaystyle y\displaystyle =\displaystyle mx+bWrite the slope-intercept form
\displaystyle y\displaystyle =\displaystyle 3x+(-5)Substitute m=3 and b=-5
\displaystyle y\displaystyle =\displaystyle 3x-5Simplify
c

What would be the surface air temperature when the water temperature is 11\degree \text{C}?

Worked Solution
Create a strategy

Substitute x=11 into the equation formed in part (b).

Apply the idea
\displaystyle y\displaystyle =\displaystyle 3x-5Write the equation
\displaystyle =\displaystyle 3\times 11-5Substitute x=11
\displaystyle =\displaystyle 28Evaluate

The surface air temperature would be 28\degree \text{C}.

d

What would be the water temperature when the surface air temperature is 31\degree \text{C}?

Worked Solution
Create a strategy

Substitute y=31 into the equation formed in part (b).

Apply the idea
\displaystyle y\displaystyle =\displaystyle 3x-5Write the equation
\displaystyle 3x-5\displaystyle =\displaystyle 31Substitute y=31
\displaystyle 3x\displaystyle =\displaystyle 36Add 5 to both sides
\displaystyle x\displaystyle =\displaystyle \dfrac{36}{3}Divide both sides by 3
\displaystyle =\displaystyle 12Evaluate

The water temperature would be 12\degree \text{C}.

Idea summary

A linear model can be used to make predictions. If we substitute in the value of one variable we can solve the linear equation to find the value of the other variable.

Outcomes

8.F.A.3

Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear.

8.F.B.4

Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.

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